TUNNELING PROPERTIES AND ELECTROMAGNETIC RESPONSE IN IRON BASED SUPERCONDUCTORS AND OTHER SYSTEMS

Abstract

An iron-based chalcogenide compound, $\text{FeTe}_{1-x}\text{Se}_{x}$ (FTS), has recently attracted attention as a potential candidate for a readily available platform for hosting the exotic topological superconducting (TSC) phase on its surface. In its 3D sample the co-existence of the strong topological insulating (TI) phase and cylindrical Fermi sheets provide the two necessary ingredients for the TSC phase on its surface - i) the topologically protected helical surface states and ii) the intrinsically induced s-wave pairing from the bulk superconductivity.

The strong TI phase in the FTS alloy is crucially dependent on the relative composition ratio between Te and Se atoms. The topological FTS sample, $\text{FeTe}_{.55}\text{Se}_{.45}$, that is widely studied for having the highest critical temperature ($T_c$) in this class, interestingly, lies close to the topological-trivial phase boundary (estimated to be close to $x=.5$ as observed by a recent experiment \cite{Brookhaven}). Furthermore, due to its alloy nature a typical sample of FTS suffers from spatial inhomogeneity in Te/Se composition at multiple length scales ranging from few nm to 100 $\mu \text{m}$. Thus, in a topological FTS sample there might be patches where the phase is driven out of its topological natures due to the local deficit of Te composition induced by Te/Se fluctuations. Such trivial domains would be scattered throughout the sample - hence, we call such a phase a topological domain disordered phase. The trivial domains in such a disordered sample would inflict inhomogeneity in the topological surface state (TSS) density distribution which we study in Chapter \ref{chap:chapter_2} of this thesis. After carefully exploring the effects of topological domain disordered phase in an effective model of FTS, we conclude that the non-topological domains on the surface are characterized by suppressed local density of states (LDOS) surrounded by ridges of enhanced LDOS throughout the energy range of the Dirac dispersion of the TSS. Moreover, the appropriately scaled LDOS at various energies within the Dirac window collapse on each other for a given disordered sample which, as we show, is in stark contrast to the case of conventional chemical potential disorder. Hence, these features are expected to appear in measurements of tunneling conductance such as in scanning tunneling spectroscopy (STS) of the FTS surface when the sample is not in the superconducting phase.

Another kind of exotic modes, namely the helical Majorana modes, appear in the 2D TSC phase at a linear defect when the TSC gap on either side differs by a phase difference of $\pi$ from the other side - thus, forming a $\pi$ shifted Josephson junction (JJ) on the TSC surface. Signatures of such helical Majorana modes have been observed by a recent tunneling experiment on the surface of FTS. In FTS such signatures are characterized by the non-zero flat density of states (DOS) in an STS measurement at a crystalline domain wall (DW) which is associated with the in-plane half-unit-cell-shift (HUCS) of the lattice. Such non-zero flat DOS within the SC gap which is consistent with the existence of linearly dispersing 1D modes is absent in the non-topological $\text{FeSe}$ sample - hinting towards the topological origin of the same in FTS. Even though the TSC phase on the surface is expected in FTS, the origin of a $\pi$ shifted order parameter that is crucial to accommodate the helical Majorana modes at the DW, is yet to be fully understood. In chapter \ref{chap:chapter_3} of this thesis we propose a mechanism that stabilizes a $\pi$-junction at the HUCS domain wall when the intrinsic superconducting pairing is of $s_{\pm}$ character as is the case for bulk FTS superconductivity that consists of hole-like pockets around $\Gamma$ and electron-like pockets around $M$ point of the Brillouin zone (BZ). We argue that if the DW induces inter-pocket transmission between the $\Gamma$ and $M$ pockets strongly enough, the coupling between the order parameters (OPs) of the two pockets ($\Gamma$ and $M$) is also enhanced. Since the $s_{\pm}$ nature of the pairing implies an intrinsic $\pi$-phase difference between the OPs associated with the $\Gamma$ and $M$ pockets, strong DW-induced coupling between them can stabilize a Josephson junction with $\pi$-phase shift. We explore the possibility of such $\pi$-junction in FTS by constructing an effective model of the Fermi surface of FTS and computing the Bogoliubov-de-Gennes (BdG) spectrum with $s_{\pm}$ pairing at the HUCS DW. Varying our model parameters within the regime of the observed FTS Fermi surface we find that for a wide range of parameters the $\pi$-junction accommodates Andreev bound states (ABSs) that have lower occupied energies than those for trivial 0-phase shifted junction. Simultaneous numerical computation of the DW-induced scattering problem reveals that the strong inter-pocket transmission strength is positively correlated with the stability of the $\pi$-junction, supporting our proposed mechanism for the $\pi$-junction stability as described earlier. Such $\pi$-Josephson junctions can in principle be detected using superconducting quantum interference device (SQUID) using either a mesoscopic device or corner junction.

In the next chapter \ref{chap:chapter_4} of the thesis we consider another setup of Josephson junction (JJ) consisting of an interacting one-dimensional quantum wire sandwiched between two semi-infinite SC leads of conventional s-wave pairing. The low energy Physics of such interacting 1D system which is known as Luttinger liquid (LL) is controlled by the two decoupled sets of eigen modes - namely, the charge and spin density waves. The SC leads in such a JJ thus allow easy access to the charge degrees of freedom in the LL - also known as plasmons which can also be probed optically using near field optical microscopy \cite{Wang2015,Wang2020}. An interesting aspect of an S-LL-S setup is that the NS interface blocks the flow of spin current due to the perfect Andreev reflection at low energies. Hence, the spin modes of the LL would typically be obscure to a transport measurement due to the aforementioned spin-charge decoupling. Hence, a conductivity measurement would not be able to detect the spin degrees of freedom if they are separated from the charge modes. However, we note that the impurity induced back-scattering processes which involve low energy but large momentum transfer can couple the spin and charge densities in the LL and we study the signatures of such spin-charge coupling in the electromagnetic (EM) response measurements. In our numerical evaluation which incorporates the impurity potential perturbatively and we calculate the resultant EM absorption spectrum using linear response theory which is measurable in a similar S-LL-S setup. We find that the signatures of back-scattering induced spin-charge coupling appear in the absorption spectrum as excitation peaks at the energies that are associated with the spin excitations of the LL. Tuning the total density of the 1D system and hence the Fermi momentum we find a regime where the spin peaks are roughly of equal amplitude as that of the charge peaks. Since in a 1D wire the density can be tuned by means of gating \cite{Wang2020}, such regime is accessible to a transport measurement. In that regime tuning the Coulomb interaction strength one can distinguish the charge peaks as they shift on the energy axis whereas the spin peaks are insensitive to such interaction strength variation. We conclude that such impurity induced signatures of spin modes can thus be probed in an S-LL-S setup by investigating the EM response of the sample in a conductivity measurement.

In the last chapter \ref{chap:chapter_5} of the thesis we go beyond the regime of linear response formalism that is discussed in relation to the EM response of the S-LL-S setup in the previous paragraph and consider the optical response up to second order in the applied EM field. A fundamental feature of non-linear response is its non-equilibrium nature which is absent in linear responses. Such non-equilibrium character is manifested as the out-of-time-ordered correlators (OTOC) in response theory associated with the second power of the applied EM field magnitude. Such correlators for a generic interacting system cannot be described using the standard Feynman diagrams, rather one requires its non-equilibrium extension which is known as the Keldysh formalism. We focus on this non-equilibrium nature of the response by considering the bulk photovoltaic effect (BPVE) in a non-centrosymmetric system where we include electron-phonon (e-ph) coupling as the mode of relaxation. Considering the semi classical limit where the e-ph scattering is stronger than the applied field strength we show how the contribution of the OTOC appears in the BPVE induced DC current measurement when the system is illuminated by an inhomogeneous profile of optical intensity similar to the case of a irradiation on a finite portion of the sample. We also introduce the Floquet master equation approach to treat the e-ph coupling quantum mechanically and demonstrate a shift in the scaling of the DC response from quadratic to linear in the limit of the e-ph coupling strength being much smaller than the applied EM field.